The group, which currently consists of Phil Collins, Tony Banks, Mike Rutherford, Daryl Stuermer, and Nic Collins, made the announcement today, saying that a further three upcoming gigs at the O2 in London would also have to be rescheduled.
The line-up does not contain original singer Peter Gabriel or guitarist Steve Hackett.
The band say that tickets for the final shows of the Last Domino? Tour will remain valid, and that the news had left the group “devastated”.
Genesis did not specify which members of the band, or even how many, had tested positive for COVID 19.
In a statement, the band said: “Following guidance and advice from the government, it is with huge regret that the final four shows of Genesis' Last Domino? Tour (this evening (8/10) at Glasgow SSE Hydro and 11, 12, 13 October at The O2 in London) have had to be postponed due to positive COVID 19 tests within the band.
“We are working to reschedule them as soon as possible and will announce the new dates through our website (www.genesis-music.com) and social media channels as soon as possible.
“All tickets will remain valid and ticket holders will be contacted by their ticket providers.
“This is a hugely frustrating development for the band who are devastated with this unlucky turn of events.
"They hate having to take these steps but the safety of the audience and touring crew has to take priority.
"They look forward to seeing you upon their return."